Theorem bd0 15554

Description: A formula equivalent to a bounded one is bounded. See also bd0r 15555. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bd0.min	⊢ BOUNDED 𝜑
bd0.maj	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
bd0	⊢ BOUNDED 𝜓

Proof of Theorem bd0

Step	Hyp	Ref	Expression
1		bd0.min	. 2 ⊢ BOUNDED 𝜑
2		bd0.maj	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	2	ax-bd0 15543	. 2 ⊢ (BOUNDED 𝜑 → BOUNDED 𝜓)
4	1, 3	ax-mp 5	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 105  BOUNDED wbd 15542

This theorem was proved from axioms:  ax-mp 5  ax-bd0 15543

This theorem is referenced by:  bd0r  15555  bdth  15561  bdnth  15564  bdnthALT  15565  bdph  15580  bdsbc  15588  bdsnss  15603  bdcint  15607  bdeqsuc  15611  bdcriota  15613  bj-axun2  15645